I thought this (ancient) paper was interesting enough to submit to HN since it's an English translation of the 1738 Latin treatise. I've seen it mentioned in a lot of places, but until tonight, I had never read the original (albeit a translation).

https://en.wikipedia.org/wiki/Utility

>"The St. Petersburg paradox was first proposed by Nicholas Bernoulli in 1713 and solved by Daniel Bernoulli in 1738. D. Bernoulli argued that the paradox could be resolved if decision-makers displayed risk aversion and argued for a logarithmic cardinal utility function. The first important use of the expected utility theory was that of John von Neumann and Oskar Morgenstern, who used the assumption of expected utility maximization in their formulation of game theory."

The St Petersburg Paradox [1] has numerous solutions. While D. Bernoulli's solution is interesting in that it introduces the concept of Utility, it isn't the most satisfying to me.

The simplest solution is to recognize that a real world casino has an upper bound to what it can pay, and you can't realistically get more than that. With that upper bound, the expected payout is now finite, and actually pretty small, even for absurdly large upper bounds on the maximum payout (such as "all the money in the world").

[1] https://en.wikipedia.org/wiki/St._Petersburg_paradox

Even if you had a 100% guarantee that the casino would pay with no upper bound, it still isn't worth paying more than few dollars for a lottery ticket.

> if you had a 100% guarantee that the casino would pay with no upper bound

That's kind of like saying "If you could travel faster than the speed of light..." Paying without an upper bound is no less a violation of the laws of physics in a finite universe.

Ha! Just this morning I was reading about the development of probability theory and risk management in "Against the Gods" by Peter Bernstein [1]. I just finished the chapter on the Bernoullis contributions. Interesting read!

[1] http://www.amazon.com/Against-Gods-Remarkable-Story-Risk/dp/...

'Against the Gods' is an excellent book. I can't recommend it highly enough to anybody who is at all interested in any kind of risk, or the financial markets.

The Physics of Wall Street covers the history of math from Bernoulli through Mandelbrot and how we arrived at modern algorithmic trading.

http://www.amazon.com/The-Physics-Wall-Street-Unpredictable/...

It doesn't cover the actual math. Not sure if there's a good book for that.

> The Physics of Wall Street covers the history of math from Bernoulli through Mandelbrot and how we arrived at modern algorithmic trading.

The way you put it makes it sound like algorithmic trading is the pinnacle achievement of probability theory. Surely not?

It's great when you can use math, and remove humans from the process once the algorithms are written, to do something people thought was impossible. Was never trying to imply that's the pinnacle. You've got Gordon Gecko and then you have James Simons.

http://youtu.be/QNznD9hMEh0

Maybe the U.S. wouldn't rank so low in math and science if kids saw a few more career possibilities.

http://www.pewresearch.org/fact-tank/2015/02/02/u-s-students...

The algo part of High frequency trading is generally overstated. The real value is the number of transactions as even really simple algorithms can make money if there are no transaction fees.

EX: Hold one stock for every cent in the current price. 1.53$ = 153 shares. Buy one stock per cent if the stock price moves down one cent and sell one stock if it moves up once cent. You now make a half cent every time the price moves up or down one cent. Note, this only works well for stocks with low share prices, lots of movement, and no transaction fees, and if the price doubles you have sold all your shares.

You have this exactly backwards: you are guaranteed to lose 0.5 cents/transaction. Suppose the price is $1.00 so you hold 100 shares. The price move up to 1.01 so you buy a share (at 1.01). Now the price moves back to 1.00 so you sell a share (at 1.00). You just lost 0.01.

There is no algorithm that guarantees you will make money in a fair market. The only way to guarantee making money is arbitrage or trading on inside information.

Ops, flipped it. If the price moves up you sell, if the price moves down you buy.

The reason this is not 'optimal' is if the price moves up to far you have sold every stock and run out, also you need a reserve to handle the price moving close to zero and if the stock goes bust you now own a lot of worthless stock.

PS: So, you will make money on a bounded random walk, but potentially far less money than holding the stock. In the end all this does is trade unbound potential gains for a finite income stream. Which is what all algo provide there effectively choosing which game to play in Vegas or shifting risk around etc.

> Ops, flipped it. If the price moves up you sell, if the price moves down you buy.

But then you aren't maintaining the invariant that you hold a number of shares equal to the current price.

> So, you will make money on a bounded random walk

Yes. If you know ahead of time what the price is going to do (even probabilistically) then you can make money. If you don't, you can't.

aren't maintaining the invariant Yes, it’s just the starting point for a really simple example.

If you know ahead of time what the price is going to do (even probabilistically) then you can make money. Exactly, you pick a model and make money ‘if’ reality fits that model. However, fast dumb models can make lots of money and complexity adds risks.

Anyway, many people get stuck with the idea you need the smartest people in the room while ignoring how much it costs to have the ‘smartest people’ in the room. Arguably, many companies are simply doing this to attract investors not because it maximizes returns as managing money is a great way to make money.

> fast dumb models can make lots of money

Sure, and they can also lose a lot of money.

> complexity adds risks

Not necessarily. Modern portfolio theory is a lot more complex than (say) buying and holding a single stock. But it's a lot less risky.

> you need the smartest people in the room

Maybe you don't need the smartest people in the room, but it helps not to have too many stupid ones. The real problem is that it can be really hard to tell which is which.

Here is a really cool TED talk with some everyday applications of Bernoulli's work:

http://www.ted.com/talks/dan_gilbert_researches_happiness?la... - Why we make bad decisions by Dan Gilbert

Not only are the ideas impressive, but the computations are also.